Modeling and Analysis of Multipath Video Transport over
Lossy Networks Using Packet-Level FEC
Xunqi Yu1 , James W. Modestino1 and, Ivan V. Bajic2
Electrical and Computer Engineering Department, University of Miami
2
School of Engineering Science, Simon Fraser University
Abstract
The use of forward error correction (FEC) coding is often proposed to combat the effects of network packet losses
for error-resilient video transmission on packet-switched networks. On the other hand, path diversity has recently
been proposed to improve network transport for both singledescription (SD) and multiple-description (MD) coded video.
In this work we model and analyze an SD coded video transmission system employing packet-level FEC in combination
with path diversity. In particular, we provide a precise analytical approach to evaluating the efficacy of path diversity in
reducing the burstiness of network packet-loss processes. We
use this approach to quantitatively demonstrate the advantages of path diversity in improving end-to-end video transport performance using packet-level FEC.
1. Introduction
To transmit packet video over lossy packet-switched networks, packet-level forward error correction (FEC) is often proposed to combat packet losses typically due to network congestion, link failures, and timeouts. The efficacy
of packet-level FEC is often limited by the bursty nature of
typical network packet-loss processes. The use of path diversity, where packets are routed over multiple paths, has
recently been proposed to improve network transport for
both single-description (SD) and multiple-description (MD)
coded video [3, 4]. Path diversity can reduce the effects of
extended packet bursts as seen at the FEC decoder, thereby
improving the FEC performance. In this work, we model
and analyze a SD-coded video transmission system using a
combination of both packet-level FEC and path diversity, and
demonstrate the efficacy of path diversity in improving joint
source-channel coding (JSCC) performance for packet video
transmission.
We consider two specific multipath transport scenarios:
In the first scenario, we simply assume the paths share no
joint links and are totally independent of each other. We provide a precise quantitative analysis of the resulting effective
loss-burst-length distribution and residual decoded packetloss rate using path diversity. Using these results, we demonstrate the efficacy of path diversity in improving end-to-end
video transmission performance using packet-level FEC. In
the second scenario, we assume different paths may share
some joint links and, therefore, the packet-loss processes on
different paths may be correlated. We investigate the effect
of the resulting path correlation on FEC performance.
The paper is organized as follows: In Section 2 we describe a general system model for packet-video transmission
over networks using packet-level FEC and path diversity. In
Section 3 we investigate the multipath video transport system for the case of disjoint paths using packet-level FEC. In
Section 4 we consider the multipath video transport system
for the case of joint paths. Finally, in Section 5 we provide a
summary and conclusions.
2. Multipath Video Transport System
Figure 1 illustrates the general SD video multipath transmission system model considered in this paper. Several
important system model parameters are also indicated. As
shown in this figure, a video transmission system has the following components: a video encoder and decoder, a packetlevel FEC encoder and decoder, and a multipath transport
network.
( R s ,ß )
( n,k )
Video
Encoder
Packet-Level
FEC Encoder
R
Path #1
Path #2
...
1
Multipath Transport
Network
Path #N
PL dec
Video
Decoder
FEC Decoder
Information Packets
Parity Packets
Figure 1. Video transmission system model.
265
L1
2.1. Video Encoder/Decoder
Assume the source generates a space-time video signal
which is used as input to the video encoder. We assume the
encoder is a typical block-based hybrid motion-compensated
video encoder 1, which encodes the video signal at rate R s
bits/sec with INTRA refresh rate β (as a fraction of macroblocks coded in the intra mode). The latter parameter is
indicative of the error resilience capability of the encoded
video. We assume the compressed video data are packetized
with M macroblocks (MBs) per transport packet.
At the video decoder, the received video packets are depacketized and the video signal is reconstructed. For lost
video MBs, passive error concealment will be used to mitigate the distortion due to unrecovered packets.
2.2. Packet-Level FEC Encoder/Decoder
In this paper we use an interlaced Reed-Solomon
RS(n, k) coding scheme [6] to provide FEC. For every block
of k information packets an additional p = n − k redundant packets are transmitted. The channel-coding rate is then
given by
bits/channel use
(1)
Rc = k/n ;
Assume the total available network bandwidth is fixed at
R bps. As a result of the overhead introduced by channel
coding, the source coding rate has to be throttled to
Rs = R ∗ Rc bps.
(2)
At the FEC decoder, packets received from different paths
will be reordered as necessary. Some packets may be lost
due to network congestion, link failures, and timeouts. Let
P (j, n) denote the block error distribution seen by the FEC
decoder after packet reordering, i.e., the probability that j
packet losses occur within a block of n consecutive packets,
n ≥ 1, 0 ≤ j ≤ n. With Np denoting the number of lost
packets within this block, if N p > n − k we assume the lost
packets within this block cannot be recovered by the FEC decoder. Then the residual packet-loss rate of the original video
packets after channel decoding can be shown to be given by
n
j ∗ P (j, n) /n .
(3)
P Ldec =
j=n−k+1
2.3. Multipath Transport Network
As illustrated in Fig. 1, the encoded video packets are
transported over the multipath transport network composed
of N paths. To simplify the analysis, we assume a simple
cyclic or round-robin multipath transport scheme: the 1-st
packet is transported on path #1, the 2-nd packet is transported on path #2, etc., until the (N + 1)-th packet which
will then be transported on path #1.
1 In Section 3.3 we will make specific use of the ITU-JVT JM 6.1 codec
for the newly developed H.264 video coding standard to provide some numerical examples.
L2
…
LK
Receiver
Sender
Figure 2. A transport path consisting of K links.
As illustrated in Fig. 2, each path consists of several transmission links connected by routers. Due to temporary buffer
overflow and link outages, the packet losses typically occur
in bursts with possibly varying burst lengths. We use a twostate discrete-time Markov-chain model, called the Gilbert
model [7], to capture the bursty nature of each link. The
Gilbert model for link i has two states, “Reception” and
i
i
and P10
, rep“Loss”, and two independent parameters: P 01
resenting the associated state transition probabilities.
In the literature, two alternative parameters are often used
to characterize the Gilbert model: the steady-state packetloss rate P Li and the average loss burst length LB i . The
relationships between these parameters are given by:
Pi
1
(4)
P Li = π i (1) = i 01 i ; LB i = i ,
P10 + P01
P10
where π i (1) is the steady-state probability of being in the
loss state.
We assume the packet-loss processes of the links along a
given path are independent. Then, the end-to-end packet loss
process of the entire path consisting of K links can be modeled as an aggregate Gilbert channel, with average packet
loss rate (P L) and average burst length (LB) given by
PL = 1 −
K
π i (0) ,
(5)
i=1
and
K
1 − i=1 π i (0)
LB = K
,
i
( i=1 π i (0))(1 − K
i=1 (1 − P01 ))
(6)
where π i (0) = 1 − π i (1) is the steady-state probability of
i
being in the reception state for intermediate link i and P 01
is
the corresponding transition probability from the reception
state to the loss state [7].
2.4. Video Distortion Models
The end-to-end distortion of a reconstructed video sequence, denoted by D, results from two components: the
distortion induced by source compression, denoted by D s ,
and the channel distortion due to packet losses, denoted by
Dc . We make use of the additivity assumption in [1], which
states that the end-to-end distortion is the sum of D s and Dc ,
i.e.,
D = Ds + Dc .
(7)
As shown in [1], the compression distortion D s can be
expressed as:
θ
+ D0 ,
(8)
Ds =
Rs − R0
where Rs is the source coding rate and θ, R 0 and D0 depend
on the INTRA rate β and other model parameters, which are
266
0
Path 1
0.06
Residual Packet−Loss Rate (PLdec)
10
N=1
Path 2
N=2
−2
10
...
Receiver
Prob {BL ≥ x}
Sender
Path N
Path 1 ( PL 1 , LB 1 )
Receiver
Sender
N=3
−4
N=4
10
N=∞
PL = 0.15
LB = 8
−6
10
...
Path 2 ( PL 2 , LB 2 )
−8
10
Path N ( PL N , LB N )
1
2
3
4
5
6
7
Loss Burst Length (x)
specific to the encoded video sequence and can be obtained
by fitting the model to experimental data.
The channel distortion (due to packet losses) D c can
be expressed as a function of the decoded packet-loss rate
P Ldec and the INTRA coding rate β as [1]
T
−1
1 − βt
,
(9)
Dc = αP Ldec
1 + γt
t=0
where T = 1/β and the parameter γ describes the efficiency
of loop filtering to remove the effects of errors due to packet
losses. Likewise, the model parameters γ and α can be obtained by fitting the model to experimental data.
3. Disjoint Paths
First, consider the case where the different paths share no
joint links so that packet-loss processes on different paths are
independent. Therefore, as illustrated in Fig. 3, each end-toend path can be modeled as an independent Gilbert channel.
The channel parameters associated with the aggregate Gilbert
channel for the i-th path, P L i and LBi , can be obtained from
(5) and (6), respectively 2.
The FEC performance is dependent on the burstiness of
the underlying packet-loss processes. Generally, the less
bursty the packet losses are, the better performance FEC can
achieve. In this subsection, we quantitatively investigate the
effectiveness of multipath transport in reducing the burstiness
of packet losses. The results are used to explain the FEC performance improvement using path diversity described in the
next subsection.
Suppose the random sequence {Y n } represents the packet
loss process perceived by an end node, with 1 denoting loss
and 0 denoting reception. The effective average loss burst
length perceived by an end node can be expressed as
k=1
2 For
length.
k ∗ P r{LBef f = k} =
∞
(k + 1) ∗ P {1k 0|01} ,
k=0
(10)
simplicity, we assume each of the disjoint paths are of the same
0.03
0.02
0.01
0
1
10
PL = 0.07
LB = 4
0.04
Bernoulli Losses
2
3
4
5
Number of Paths (N)
Figure 5. Residual packet-loss rates
P Ldec vs. number of paths N .
where the random variable LB ef f denotes the loss-burstlength seen by the receiver after packet reordering.
Here we model each of the N independent paths as an aggregate Gilbert model. To simplify the analysis, we assume
these Gilbert models are homogeneous, each with average
packet-loss probability and average burst length P L and LB,
respectively. Furthermore, we assume the packets are transmitted over the network using the cyclic multipath transport
scheme described in Section 2.3. Then the loss-burst-length
distribution, P r{LBef f = k + 1}, k ≥ 0, can be expressed
as

k
 P L (1 − P L)
k
P {1 0|01} =
P LN−2 P00

k−N+1
P LN−2 P01 P11
P10
;
;
;
0≤k ≤N −3
k =N −2
k ≥N −1,
(11)
where P01 , P10 are the state transition probabilities of each of
the aggregate Gilbert models and can be computed from P L
and LB according to (4) with P 00 = 1 − P01 , P11 = 1 − P10 .
Therefore, from (10) and (11), the effective average loss
burst length is
ALBef f
3.1. Analysis of Loss-Burst-Length Statistics
∞
9
Figure 4. Complementary cdf of
loss-burst-length for different N .
Figure 3. Multipath transport
network with disjoint paths.
ALBef f =
8
RS (7,5)
RS (15,9)
0.05
=
=
+
+
=
∞
(k + 1)P {1k 0|01}
k=0
N−3
(k + 1)P Lk (1 − P L)
k=0
N−2
(k + 1)P LN−2 P00
k=N−2
∞
k−N+1
N−2
P01 P11
P10
N−1 P L
1/(1 − P L) .
(12)
This last expression indicates that, somewhat surprisingly,
when the number of paths N ≥ 2, the effective average
loss-burst-length ALB ef f is independent of N and LB, but
only depends on P L. Figure 4 provides a numerical example of the complementary cdf of the loss-burst-length,
P r{LBef f ≥ x}, with different orders of path diversity
N . For comparison, we indicate the corresponding complementary cdf for the Bernoulli channel (N = ∞), where the
losses are totally independent 3. It indicates that, although the
effective average loss-burst-length ALB ef f is the same for
N ≥ 2, the loss-burst-length distributions are quite different. More specifically, it shows that, with an increase of path
3 When N → ∞, each packet will be transmitted on a different path.
In this case, the channel reduces to a Bernoulli channel, where the packet
losses are totally independent.
267
N
RS(n,9)
N=∞
32
30
N=5
Path 1
Path 2
24
Receiver
...
Sender
N=1
26
Independent paths
Path N
22
Dj. Path 1 ( PL 1 , LB 1 )
No FEC
20
18
0
Receiver
Sender
28
J. Path (PL J , LB J)
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Sender
Dj. Path 2 ( PL 2 , LB 2)
...
End−to−end Performance (in PSNR)
34
Packet−Loss Rate (PL)
Figure 6. The end-to-end PSNR perfor-
3.2. Analysis of Residual Packet-Loss Rates
We need to obtain the block error distribution P (j, n) to
evaluate P Ldec according to (3). Consider an arbitrary block
of n packets. Without loss of generality, we assume the 1-st
packet within this block is transmitted on path #1. Assume
there are N paths in total. Then, out of n packets, the number
of packets transmitted on the i-th path is
n
+ hi ,
1≤i≤N,
(13)
li =
N
where
hi =
1,
0,
if i ≤ n mod N ,
otherwise .
(14)
Out of these n packets, let b i denote the number of lost packets transmitted on the i-th path. The total number of lost
packets is then
N
bi = j .
(15)
i=1
Since the packet-loss processes of these N path are independent of each other, we have
N
Pi (bi , li ) ,
(16)
P (j, n) =
S
i=1
where the sum is over the set
S=
(b1 , b2 , . . . , bN ) :
N
bi = j
Correlated paths with 2 joint links
Figure 7. Multipath transport network with joint paths.
diversity order N , the probability P r{LB ef f ≥ x} will decrease for relatively large x. This means that with increasing
path diversity N , the residual packet-loss probability with
FEC, P Ldec , will be reduced. We will further demonstrate
this in the next subsection.
,
(17)
i=1
and Pi (bi , li ) denotes the block error distribution on the i-th
path. If we model the packet-loss process over each path as
Receiver
Receiver
Dj. Path N ( PL N , LB N)
mance vs. packet-loss rate P L for the Susie
sequence with RS(n, 9) codes and path diversity N , where the FEC coding rates Rc
are optimally selected; Other model parameters are set as follows: R = 80 Kbps,
β = 0.02, LB = 4.
Sender
Figure 8. Path correlation
model.
a Gilbert model, as expressed by (5) and (6), then the block
error distribution P i (bi , li ) on each path can be obtained by a
recursive algorithm first proposed in [8].
Figure 5 provides a numerical example of the efficacy of
path diversity in reducing the residual packet-loss rates for
the RS(7, 5) and RS(15, 9) codes, where each path is modeled as a homogeneous and independent Gilbert model with
P L = 0.07 and LB = 4. It demonstrates that, as expected,
packet transport with path diversity can improve the efficacy
of FEC coding significantly. We have also indicated in Fig. 5
the limiting residual packet loss performance if the losses are
independent (N → ∞) 4 , i.e., a Bernoulli channel. Observe
the rapid approach of the residual packet-loss rates to their
limiting values with increasing N . Also note that for use of
a fixed code there is little advantage to path diversity orders
N > 4. Additional results described in [9] also demonstrate
the effectiveness of multipath transport in reducing the probability of large bursts and their effect on end-to-end performance.
3.3. Video Performance Using Multipath Transport
Figure 6 demonstrates a comparison of the end-to-end
video performance with different network packet-loss rates
(P L) for the following cases: 1) without coding (R c = 1)
or path diversity (N = 1); 2) with an RS(n, 9) code, but no
path diversity (N = 1), where R c is optimally selected5 ; 3)
combined RS(n, 9) code and path diversity (N ≥ 2), where
Rc is again optimally selected, for the QCIF Susie test sequence. For comparison, we have also indicated the performance achieved on a Bernoulli channel (N = ∞). The QCIF
Susie sequence (176×144) consists of 150 frames at f r = 30
4 The limiting value for the RS(15, 9) code is extremely small and indistinguishable from zero in Fig. 5.
5 Optimally selecting R to maximize end-to-end performance for a
c
given overall transmission rate R represents a joint source-channel coding
(JSCC) approach.
268
frames/sec. We assume every M = 11 MBs are packetized
into one packet. Therefore, each QCIF frame is packetized
into 99/M = 9 packets. Other system parameters are indicated in the figure caption. This figure indicates that, compared to the case of JSCC without path diversity, the combination of JSCC and path diversity can provide significantly
improved end-to-end video performance. For example, for
P L = 15%, there is a performance advantage of 2 dB in
going from N = 1 to N = 4. However, unlike the fixed
code case illustrated in Fig. 5, observe that when the code
rate is optimally chosen as part of a JSCC approach there is a
considerable performance advantage to path diversity orders
N > 4.
4. Correlated Paths
In the previous section, we assumed that the different
paths share no joint links so that the packet-loss processes on
different paths are independent. However, in actual multipath
transport networks there may be some shared or joint links
between different transport paths. In this case, the packetloss processes on different paths may be correlated. In actual
networks the connection topologies (joint/disjoint links) between the sender and the receiver may be quite varied so that
precise modeling of video transport using path diversity can
be fairly complex. However, it has been shown in [4, 5] that
the important end-to-end properties of a 2-path network can
be captured using a simplified three-subpath topology, where
subpaths 1 and 2 are formed by the disjoint links along the
two paths and subpath 3 is formed by the joint links along
both paths. In this section we consider a similar approach as
in [4, 5], except that we will consider a more general N -path
network. We will specifically concentrate on the effect of
path correlation on the end-to-end video performance using
packet-level FEC. To model the effect of path correlation on
FEC performance, we consider a simplified scenario: different paths share common joint links, as shown in the upper
subfigure of Fig. 7, i.e., a number of disjoint links followed
by a series of joint links.
4.1. Analysis of Residual Packet-loss Rates
We assume each subpath can be described by a Gilbert
model, as illustrated in the bottom subfigure of Fig. 7, where
the associated model parameters can be derived from the corresponding portion of the original path using (5) and (6). To
evaluate the residual packet-loss rates P L dec after FEC decoding, again we need to determine the block error distribution P (j, n) as seen by the FEC decoder after packet reordering. Assume there are j packets lost out of n consecutive packets. If the total number of packets lost over disjoint
links is bD = d then the number of packets lost over the joint
links is bJ = j − d. Therefore, we have
j P r{bD = d} ∗ P r{bJ = j − d} . (18)
P (j, n) =
d=0
Since out of these n packets d packets have already been lost
over the disjoint links, only the remaining n − d packets will
be transmitted over the joint links. Therefore, we have
P r{bJ = j − d} = P J (j − d, n − d) ,
(19)
where P J (j − d, n − d) denotes the block error distribution
over the joint path.
Let bD
i denote the number of packets lost over disjoint
path i. Therefore, we have
N
bD
(20)
bD =
i .
i=0
Following a similar approach as in going from (15) to (17),
it can be shown that [9]
P (j, n) =
j N
d=0
Sd
PiD (bD
i , li )
∗ P J (j − d, n − d) ,
i=1
(21)
where li is given by (13) and
N
D D
D
D
Sd =
b 1 , b 2 , . . . , bN :
bi = d .
(22)
i=1
If we model the packet-loss process over each
joint/disjoint subpath as a Gilbert model, then the block
J
error distributions P iD (bD
i , li ) and P (j − d, n − d) on each
of the joint/disjoint paths can be obtained by the recursive
algorithm originally proposed in [8].
4.2. Effect of Path Correlation on Packet-Level FEC
Performance
In order to investigate the effect of path correlation on
FEC performance, we consider a simplified 2-path transport
scheme and assume there are 5 intermediate links on each of
the 2 paths, as illustrated in Fig. 8. We further assume that
each intermediate link is independently modeled by a Gilbert
channel with parameters P L i and LB i . Therefore, the corresponding end-to-end P L and LB can be computed from (5)
and (6), respectively, for each path. However, the two paths
may share some joint links. Let J denote the number of joint
links. The larger J is, the more correlated the two paths are.
Figure 8 shows the case of J = 0 and J = 2. The block
packet-loss distribution P (j, n) and the residual packet-loss
rate after channel decoding P L dec can be obtained from (21)
and (3), respectively.
Table 1 shows the effect of path correlation on the FEC
performance with two different RS codes and for two channel
conditions. More specifically, it shows the residual packetloss rates P Ldec using a relatively weak RS(22, 18) code
and a stronger RS(31, 18) code under two different channel
conditions: a relatively bad channel with P L = 10% and
LB = 8 and a relatively good channel with P L = 5% and
LB = 4. Generally, the results indicate that increased correlation among paths results in a higher residual packet-loss
rate P Ldec after channel decoding. However, it should be
269
32
J
0
1
2
3
4
5
P L = 10%, LB = 8
RS(22, 18) RS(31, 18)
8.05 %
3.03 %
8.12 %
3.09 %
8.20 %
3.34 %
8.28 %
3.58 %
8.37 %
3.80 %
8.46 %
4.01 %
End−to−end Performance (in PSNR)
noted that, when the channel conditions are relatively good,
or when a weak code is used, the path correlation level (J)
does not make much difference in the FEC performance. But
when the channel conditions are relatively bad (P L = 10%
and LB = 8) and a stronger code is used, the path correlation
level J can have a significant effect on the FEC performance.
This means that path correlation has a significant impact on
FEC performance only when channel conditions are severe
and strong FEC codes are used. These analytical results are
consistent with the conclusions in [2], where the results there
were obtained by simulations.
30
N=∞
N
N=5
RS(n,9)
29
28
27 N = 1
26
25
No FEC
24
23
0
P L = 5%, LB = 4
RS(22, 18) RS(31, 18)
2.84 %
0.32 %
2.90 %
0.34 %
2.95 %
0.39 %
3.00 %
0.43 %
3.05 %
0.46 %
3.11 %
0.49 %
Table 1. Decoded packet-loss rates (P L dec) of a 2path transport network with different path correlations
and FEC code schemes.
31
1
2
3
4
5
Number of Joint Links (J)
Figure 9. The end-to-end PSNR performance vs. the
number of joint links J for the Susie sequence with
RS(n, 9) codes and path diversity N , where the FEC
coding rates Rc are optimally selected; Other model
parameters are set as follows: R = 80 Kbps, β = 0.02,
P L = 15%, LB = 4.
strated the effect of path correlation on the packet-level FEC
performance.
References
4.3. Effect of Path Correlation on Video Performance
Figure 9 demonstrates the effect of path correlation on
video performance transmitted over different multipath transport networks. Specifically, it demonstrates a comparison of
the end-to-end video performance achieved over multipath
transport networks with different numbers of joint links (J)
for different path diversity orders (N ) and coding strategies
(with or without coding) as shown in Fig. 6. The path correlation model is the same as described in Section 4.2, but
with different path diversity N . For the coded systems, the
code rate Rc is chosen optimally. Other model parameters
are indicated in the caption. This figure indicates that, for
all path correlation levels (J), increased path diversity combined with JSCC generally can provide improved end-to-end
video performance. However, with increased path correlation, the advantage achieved with the use of path diversity
will decrease, and with J ≥ 4 there is little difference in
performance independent of the value of N .
5. Summary and Conclusions
We have modeled and analyzed an SD coded video transmission system employing packet-level FEC in combination
with path diversity. We provided a precise analytical approach to evaluating the efficacy of path diversity in reducing the burstiness of packet-loss processes. Using this approach we have quantitatively demonstrated the advantages
of path diversity in improving end-to-end video transport using packet-level FEC. Finally, we have quantitatively demon-
[1] K. Stuhlmuller, et al., “Analysis of video transmission
over lossy channels,” IEEE J. Select. Areas in Comm.,
vol.18, pp. 1012-1032, June 2000.
[2] Q. Qu et al., “On the effects of path correlation in multipath video communications using FEC,” IEEE Globecom, pp. 977-981, Nov. 2004.
[3] J. G. Apostolopoulos, “Reliable video communication
over lossy packet networks using multiple state encoding
and path diversity,” VCIP, pp. 392-409, San Jose, CA,
Jan. 2001.
[4] J. G. Apostolopoulos and M. D. Trott, “Path diversity
for enhanced media streaming,” IEEE Communications
Magazine, pp. 80-87, Aug. 2004.
[5] J. G. Apostolopoulos et al. “Modeling path diversity
for multiple description video communication,” ICASSP,
pp. 2161-64, May 2002.
[6] V. Parthasarathy, J.W. Modestino, and K.S. Vastola, “Reliable transmission of high-quality video over ATM networks,” IEEE Trans. on Image Proc., vol.8, pp. 361-374,
Mar. 1999.
[7] E. N. Gilbert, “Capacity of a burst-noise channel,” Bell
Sys. Tech. Journal, vol.39, pp. 1253-1266, Sept. 1960.
[8] E. O. Elliott, “A model of the switched telephone network for data communications,” Bell Sys. Tech. Journal,
vol.44, pp. 89-109, Jan. 1963.
[9] Xunqi Yu, “Video transmission on packet-switched networks,” Ph.D. dissertation, Department of Electrical
and Computer Engineering, University of Miami, Coral
Gables, FL 33124, in progress.
270